We show that the free hyperplane arrangements form a Zariski-closed set in various parameter spaces. Given a geometric lattice L let V(L) be the parameter space of arrangements with intersection lattice isomorphic to L. Coupling our result with a theorem of Yuzvinsky, we conclude that in V(L) the free arrangements are parameterized by a union of connected components of V(L).

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S × S.

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A ∪ {n}) = 1. The number of nonempty subsets of A which are relatively prime to n is Φ(A,n) and the number of such subsets of cardinality k is Φk(A,n). Given positive integers l1, l2, m2, and n such that l1 ≤ l2 ≤ m2 we give Φ([1,m1] ∪ [l2,m2], n) along with Φk([1,m1] ∪ [l2,m2],...

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

This report defines various operations and describes algorithms for weighted multi-tape automata (WMTAs). It presents, among others, a new approach to multi-tape intersection, meaning the intersection of a number of tapes of one WMTA with the same number of tapes of another WMTA, which can be seen as a generalization of transducer intersection. In our approach, multi-tape intersection is not co...